The conjugate gradient methods for solving the generalized periodic Sylvester matrix equations

Sun, Mingxuan; Wang, Yiju

doi:10.1007/s12190-018-01220-3

Cited by 11 publications

(6 citation statements)

References 35 publications

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“…Due to the important role that the time-varying Lyapunov equation plays in a broad spectrum of areas, there has been a rapid increase in its algorithm design, and many numerical methods and neural dynamics have been proposed to solve this problem and its timeinvariant version; see, e.g., [1][2][3][4][5][6] on this subject. Let t 0 ∈ R and t f ∈ R denote the start and the final time instant of the solving process, respectively.…”

Section: Introductionmentioning

confidence: 99%

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

Sun

Liu

2020

Adv Differ Equ

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The Zhang neural network (ZNN) has become a benchmark solver for various time-varying problems solving. In this paper, leveraging a novel design formula, a noise-tolerant continuous-time ZNN (NTCTZNN) model is deliberately developed and analyzed for a time-varying Lyapunov equation, which inherits the exponential convergence rate of the classical CTZNN in a noiseless environment. Theoretical results show that for a time-varying Lyapunov equation with constant noise or time-varying linear noise, the proposed NTCTZNN model is convergent, no matter how large the noise is. For a time-varying Lyapunov equation with quadratic noise, the proposed NTCTZNN model converges to a constant which is reciprocal to the design parameter. These results indicate that the proposed NTCTZNN model has a stronger anti-noise capability than the traditional CTZNN. Beyond that, for potential digital hardware realization, the discrete-time version of the NTCTZNN model (NTDTZNN) is proposed on the basis of the Euler forward difference. Lastly, the efficacy and accuracy of the proposed NTCTZNN and NTDTZNN models are illustrated by some numerical examples.

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Section: Introductionmentioning

confidence: 99%

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

Sun

Liu

2020

Adv Differ Equ

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“…Matrix equations are often encountered in control theory [1,2], system theory [3,4], and stability analysis [5][6][7]. For example, the stability of the autonomous systemẋ(t) = Ax(t) is determined by whether the associated Lyapunov equation XA + A X = -M has a positive definite solution X, where M is a given positive definite matrix with approximate size [8].…”

Section: Introductionmentioning

confidence: 99%

“…To solve equation (1.1) or its special cases or generalized versions, different methods have been developed in the literature [5,7,[9][10][11][12][13][14][15][16][17][18][19], which belong to the category of iterative methods. For example, two conjugate gradient methods are proposed in [7] to solve consistent or inconsistent equation (1.1). Both have finite termination property in the absence of round-off errors and can get least Frobenius norm solution or least-squares solution with the least Frobenius norm of equation (1.1) when they adopt some special kind of initial matrix.…”

Section: Introductionmentioning

confidence: 99%

Two modified least-squares iterative algorithms for the Lyapunov matrix equations

2019

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In this paper two modified least-squares iterative algorithms are presented for solving the Lyapunov matrix equations. The first algorithm is based on the hierarchical identification principle, which can be viewed as a surrogate of the least-squares iterative algorithm proposed by Ding et al., whose convergence has not been proved until now. The second one is motivated by a new form of fixed point iterative scheme. With the tool of a new matrix norm, the proof of both algorithms' global convergence is offered. Furthermore, the feasible sets of their convergence factors are analyzed. Finally, a numerical example is presented to illustrate the rationality of theoretical results.

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“…which again includes the well-known Lyapunov matrix equations and the Stein matrix equations as its special cases [5][6][7]. The SMEs serves as a basic model arising from control theory, system theory, stability analysis etc.…”

Section: Introductionmentioning

confidence: 99%

Noise-tolerant continuous-time Zhang neural networks for time-varying Sylvester tensor equations

Sun

Liu

2019

Adv Differ Equ

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In this paper, to solve the time-varying Sylvester tensor equations (TVSTEs) with noise, we will design three noise-tolerant continuous-time Zhang neural networks (NTCTZNNs), termed NTCTZNN1, NTCTZNN2, NTCTZNN3, respectively. The most important characteristic of these neural networks is that they make full use of the time-derivative information of the TVSTEs’ coefficients. Theoretical analyses show that no matter how large the unknown noise is, the residual error generated by NTCTZNN2 converges globally to zero. Meanwhile, as long as the design parameter is large enough, the residual errors generated by NTCTZNN1 and NTCTZNN3 can be arbitrarily small. For comparison, the gradient-based neural network (GNN) is also presented and analyzed to solve TVSTEs. Numerical examples and results demonstrate the efficacy and superiority of the proposed neural networks.

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The conjugate gradient methods for solving the generalized periodic Sylvester matrix equations

Cited by 11 publications

References 35 publications

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

Two modified least-squares iterative algorithms for the Lyapunov matrix equations

Noise-tolerant continuous-time Zhang neural networks for time-varying Sylvester tensor equations

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